I don’t understand what problem they are meant to solve. If you have a FOSS piece of software, you can install it via the package manager. Or the store, which is just a frontend for the package manager. I see that they are distribution-independent, but the distro maintainers likely already know what’s compatible and what your system needs to install the software. You enjoy that benefit only through the package manager.
If your distro ships broken software because of dependency problems, you don’t need a tool like Flatpak, you need a new distro.
It is a portable binary distribution format. It helps developers provide a format that should run on any Linux system and it helps users to not have to build software from source. Keep on mind building software for a specific distribution is painful. One has to build a version for every distribution and version out there. Not something a developer can really do. So without a common binary format a developer is stuck providing source archives and maybe a binary package for a few distributions they test against.
This is primarily useful from a user point of view when you want something not in your distributions repo. Either newer or just not there. If you use a Debian based distribution this does not happen often as their repo is huge though often older. Not Debian based, then it is a bigger concern. I used Redhat Desktop 25 years ago … did a lot of building from source back then. One reason I switched to Ubuntu (a Debian based distribution) at the time, huge repo.
These binary package formats also offer other features too like more isolation but they are less integrated then native packages so frankly I do not like them very much. They also increase your direct facing supply chain and do not benefit from audit and patching from your distributions security team. So again not great.